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\centerline{\bfb Reversibility in Infinite Hamiltonian Systems}
\centerline{\bfb with Conservative Noise}
\vskip.5cm
\centerline{\bf In Memoriam Roland Dobrushin}
\vskip1cm
\centerline{{\rmb J\'ozsef Fritz}$^1$,
{\rmb Carlangelo Liverani}$^2$ and {\rmb Stefano Olla}$^3$}
\footnote{}{$^1$~Department of Probability and Statistics,
E\"otv\"os Lor\'and University of Sciences,
H-1088 Budapest, M\'uzeum krt. 6-8, Hungary.
E-mail: jofri@cs.elte.hu}
\footnote{}{$^2$~ II Universit\'a di Roma ``Tor Vergata'',
Dipartimento di Matematica, 00133 Roma, Italy.
E-mail: liverani@mat.utovrm.it}
\footnote{}{$^3$~ Universit\'e de Cergy--Pontoise, D\'epartement de
Math\'ematiques, 2 avenue Adolphe Chauvin, Pontoise 95302 Cergy--Pontoise
Cedex, France and Centre de Math\'ematiques Appliqu\'ees, Ecole Polytechnique,
91128 Palaiseau Cedex, France.\hfill\break
E-Mail: olla@paris.polytechnique.fr}
\footnote{}{\bf\noindent Work partially supported by grants CIPA-CT92-4016 and
CHRX-CT94-0460 of
the Commission of the European Community, and by grant T 16665 of
the Hungarian NSF. Two of us (J.F. and C.L.) acknowledge hospitality of Ervin
Schr\"odinger Institute.}
\vskip.5cm
\noindent{\bf Abstract:} {\sl The set of stationary measures of an infinite
Hamiltonian system with noise is investigated. The model consists of
particles moving in $\RR^3$ with bounded velocities and subject to a
noise that does not
violate the classical laws of conservation, see [OVY]. Following
[LO] we assume that the noise has also a finite radius of interaction,
and prove that translation invariant stationary states of finite
specific entropy are reversible with respect to the stochastic
component of the evolution. Therefore the results of { [LO]} imply
that such invariant measures are superpositions of Gibbs states.}
\vskip.5cm
{\bf 0. INTRODUCTION}
\vskip .25cm
\numsec=0\numfor=1\numtheo=1
Let $\O$ denote the space of locally finite
configurations $\o=(q_\a,p_a)_{\a\in I}$ indexed by a countable set
$I\,,$ that is $q_\a,p_\a\in\RR^3$ are the position and momentum
of particle $\a\in I\,;$ the set $\{q_\a\}_{\a\in I}$ has no limit
points in $\RR^3$ by assumption. The classical
dynamics of the system is governed by a formal Hamiltonian $\Cal H\,,$
$$
{\cal H}(\o)=\sum_{\a\in I}\phi(p_\a)+{1\over2}
\sum_{\a\in I}\sum_{\b\ne\a}V(q_\a-q_\b)\,,
$$
where the kinetic energy $\phi:\RR^3\mapsto\RR$ is strictly convex with
bounded derivatives, and $V:\RR^3\mapsto\RR$ is a symmetric and superstable
pair potential of finite range. The associated Liouville operator will
be denoted by $L\,,$
$$
L\p=\sum_{\a\in I}\quad\bigl\lan{\dd\Cal H\over\dd p_\a},{\dd\p\over\dd
q_\a}\bigr\ran
-\bigl\lan{\dd\Cal H\over\dd q_\a},{\dd\p\over\dd p_\a}\bigr\ran\,,
$$
where $\lan\cdot,\cdot\ran$ denotes the usual scalar product in
$\RR^3$. Almost nothing is known on the ergodic properties of such infinite
systems. In fact, very few results are available even for finite systems
of this type (e.g., [KSS], [DL], [BLPS], [LW]). To ensure a proper ergodic
behavior of the system we
add some noise whereby obtaining stochastic equations of motion; these
equations read as
$$
\eqalign{
dq_\a&=\phi'(p_\a)\,dt\,,\cr
dp_\a&=-\sum_{\b\neq\a}V'(q_\a-q_\b)\,dt
+b_\a(\o)\,dt+\sum_{\t=1}^d\sum_{\b\neq\a}
\sigma^\t_{\a,\b}(\o)\,dw^\t_{\a,\b} ,}
\Eq(stochdyn)
$$
where $w^\t_{\a,\b}$ is a family of independent one-dimensional Wiener
processes for $\t=1,2,...,d$ and $\a\ne\b$ such that
$w^\t_{\a,\b}=-w^\t_{\b,\a}\,;$ $\phi'$ and $V'$ denote the gradient of
$\phi$ and $V\,,$
respectively. The coefficients $b_\a,\sigma^\t_{\a,\b}:\O\mapsto\RR^3$
are smooth local functions to be specified in the next section in such
a way that total energy and momentum are both preserved by the
randomized evolution (0.1). In addition, any Gibbs state $\PP$
with energy $\Cal H$ will be a reversible measure for the stochastic
part of the evolution:
$$
\int \p(\o)\wh L\psi(\o)\,\PP(d\o)=\int\psi(\o)\wh L\p(\o)\,\PP(d\o)
\Eq(reversibility)$$
for all smooth local functions $\p,\psi:\O\mapsto\RR\,,$ where
$$
\wh L\psi=\sum_{\a\in I} \bigl\lan b_\a,{\dd\psi\over\dd p_\a}\bigr\ran
+{1\over4}\sum_{\t=1}^d\sum_{\a\in I}\sum_{\b\ne\a}
\bigl\lan\sigma^\t_{\a,\b},(D^2_{\a,\b}\psi)\sigma^\t_{\a,\b}\bigr\ran\,,
\Eq(stochgen)$$
and $D^2_{\a,\b}\psi$ is the matrix of second derivatives obtained
by applying $D_{\a,\b}=\dd/\dd_{p_\a}-\dd/\dd_{p_\b}$ twice to
$\psi\,.$ Since the Liouville operator is antisymmetric with respect to
Gibbs distributions, the full generator, $\wt L=L+\wh L$ also
satisfies the stationary Kolmogorov equation, $\smallint \wt
L\psi\,d\PP=0$ for a wide class of test functions $\psi$ and any Gibbs
state $\PP\,.$
The converse problem is much more complex. In our basic reference [LO]
it is shown that if a translation invariant measure $Q$ with finite
specific entropy satisfies the stationary Kolmogorov equation and (0.2),
together with some other technical conditions, then $Q$ enjoys
the Gibbs property. Let us remark that finiteness of
specific entropy is a fairly natural and effective condition in the
theory of hydrodynamics limits (see [OVY]). On the contrary,
condition (0.2)
looks rather restrictive and, at least in general, not particularly
natural. The main purpose of this paper
is to show that condition (0.2) of reversibility is superfluous (i.e., it
follows from the stationarity of the measure).\nfootnote{Note that in
applications to hydrodynamics the reversibility (0.2) is insured by
construction (see [OVY] lemma 4.4), hence the present paper does not add to
hydrodynamics type problems for which the results in [LO] suffice. The
focus here is on the classification of stationary translation invariant
measures.} To obtain such a result we are forced to prove the
existence of a semigroup defined by (0.1);
its regularity (locality) will play a crucial role in the argument. This
problem may not seem to be a very difficult one since $\phi'(p_\a)\,,$
the velocity of particle $\a$, is bounded by assumption. However,
the evolution must be defined for a very large set $\bar\O\subset\O$ of
initial configurations: we need
$Q(\bar\O)=1$ for any probability measure $Q$ of finite specific entropy.
On the other hand, to obtain the necessary regularity properties of the
dynamics we have to restrict the configuration space by excluding
extremely high values of particle density.
We shall see that the desired construction fails unless the dimension
of the space is less than four, cf. {[FD]} and {[S]}.
\vskip .5cm
{\bf 1. NOTATIONS AND RESULTS}
\vskip.25cm
\numsec=1\numfor=1\numtheo=1
Configurations can be interpreted as $\sigma$-finite integer valued
measures on $\RR^3\times\RR^3\,;$ sometimes we write $\o=(q,p)$ with
$q=(q_\a)_{\a\in I}$ and $p=(p_\a)_{\a\in I}\,,$ and if
$\L\subset\RR^3$ then $\o_\L$ denotes the restriction of $\o$ to
$\L\,,$ i.e. $\o_\L=(q_\a,p_\a)_{q_a\in\L}\,,$ while $|\o_\L|$ is the
cardinality of this set. The centered cubic box of side $r>0$ will be
denoted by $\Lambda_r\,.$ Referring to functionals
$\o(\p):=\sum_{\a\in I} \p(q_\a,p_\a)\,,$
where $\p:\RR^3\times\RR^3\mapsto\RR$ is continuous with compact
support, we equip $\O$ with the associated weak topology and Borel
structure, and $C_0(\Omega)$ denotes the space of cylinder (local)
functions
$\Psi(\o)=f(\o(\p_1),\o(\p_2),...,\o(\p_n))$ such that $f\in
C(\RR^n)\,.$
Since all sets $\Sigma(\delta)$, defined for an increasing sequence
$\delta=(\delta_1,\delta_2,...)$ such that $\delta_1\ge 1$ by
$$
\Sigma(\delta):=\bigl\{\o\in\O\,:\,|\o_{\Lambda_n}|\le\delta_n
\hbox{ and } |p_\a|\le\delta_n\hbox{ if } q_\a\in\Lambda_n\bigr\}
$$
are compact, we need not worry too much
about topology. Indeed, in the forthcoming considerations we always do
have an a priori bound allowing us to restrict calculations to some
compact $\Sigma(\delta)\,.$ In this situations all reasonable
topologies coincide, moreover any continuous function can be uniformly
approximated by elements of $C_0(\Omega)$ in view of the
Stone-Weierstrass Theorem.
\vskip 5pt
\noindent{\bf Interaction}
\vskip 2pt
We consider a repelling pair potential $V$ of finite range such that
$V(x)=V(-x)$ is twice continuously differentiable, $V(0)>0$ but
$V(x)=0$ if $|x|>R_0\,,$ finally $\lan x,V'(x)\bigr\ran\le0$ for all
$x\in\RR^3\,.$ These conditions imply that $V$ is superstable, see [R]:
for each
cubic box or ball $\L\subset\RR^3$ there exist some constants $A_\L\ge0$
and $B_\L>0$ such that, for any configuration, we have
$$
\sum_{\a:q_\a\in\Lambda}\sum_{\b\ne\a} V(q_\a-q_\b)
\ge B_\L |\o_\Lambda|^2- A_\L |\o_\Lambda| .
\Eq (superstability)
$$
\vskip 5pt
\noindent{\bf Kinetic energy}\vskip 2pt
We assume that $\phi$ has bounded second derivatives, and velocities
are also bounded, i.e. $|\phi'(y)|\le \bar c<+\infty$ for all
$y\in\RR^3\,.$ To define Gibbs measures we need a lower bound:
$\lim\inf_{|y|\to\infty}\phi(y)/|y|\ge \underline c>0\,.$ When results
of [LO] are applied and extra technical condition on $\phi$ is needed.
For simplicity one can consider the case in which $\phi (y)=\sum_{i=1}^3
\phi_0(y_i)$ for $y=(y_1,y_2,y_3)\,,$ where $\phi_0\in
C^{\infty}(\RR)$ is strictly convex and
$$
{1\over 2}{d^2\over du^2}(\phi_0''(u))^2=
\phi_0'''(u)^2+\phi_0^{iv}(u)\phi''(u)\neq 0
$$
apart from, at most, finitely many points (see [LO], section two,
``Condition on the Noise" for the general condition that $\phi$
must satisfy).
Notice that if ${d^2\over du^2}(\phi''_0(u))^2= 0$ for each $u$ and if
we require the natural condition $\phi_0(u)=\phi_0(-u)$ then
$\phi_0''$ is a constant, which is the classical case of a quadratic
kinetic energy function.
\vskip 5pt
\noindent{\bf Stochastic perturbation of classical dynamics}
\vskip 2pt
There are several ways to select the coefficients of the stochastic
perturbation, we set
$$
b_\a(\o)=\sum_{\b\ne\a}\gamma_{\a,\b}(q)F(p_\a,p_\b)\hbox{ and }
\sigma^\t_{\a,\b}(\o)=\sqrt{\gamma_{\a\b}(q)}G_\theta(p_\a,\,p_\b)\,,
\Eq (coeff)$$
where, as in { [LO]},\nfootnote{In fact, in [LO], the functions
$\gamma_{\a\b}$
depend only on the variables $q_\a$ and $q_\b$; yet all is done
there applies
without changes to the situation described here.}
$\ga_{\a,\b}(q)=\ga_{\b,\a}(q)\ge 0$ is
continuously differentiable, $\gamma_{\a,\b}(q)>0$ if
$|q_\a-q_\b|< R_1$ and it is zero for $|q_\a-q_\b|\ge R_1\,,$ i.e.
$R_1>R_0$ is the radius of stochastic interaction. The functions
$F,G_\t:\RR^6\mapsto \RR^3$ are infinitely
differentiable and bounded together with their derivatives; they
are chosen in such a way that the stochastic interaction also
preserves the total momentum and energy of an interacting
couple of particles. Moreover, $\{G_\theta\}_{\theta=1}^d$ spans,
at each point, all $\RR^3$. It is natural to assume that $\ga_{\a,\b}$
depends only on the interparticle distances, and it does not depend on a
coordinate $q_\de$ if $|q_\a-q_\de|>R_2$ or $|q_\b-q_\de|>R_2\,,$ where
$R_2>2R_1$ is a constant. Therefore the stochastic interaction is also
translation invariant and has a finite range $R_3:=R_1+R_2\,.$ A new
feature of the present model is that
$\gamma_{\a,\b}$ vanishes when the number of particles near $q_\a$ or
$q_\b$ tends to infinity. For convenience, we set
$\ga_{\a,\b}(q)=\si(q_\a-q_\b)\Te_{\a,\b}(q)\,,$ where
$$
\Te_{\a,\b}(q):=\Bigl(1+\sum_{\de\in I}\chi(q_\a-q_\de)
+\sum_{\de\in I}\chi(q_\b-q_\de)\Bigr)^{-1} .
\Eq(gamma)$$
In (1.3) $\sigma,\chi:\RR^3\mapsto[0,1]$ are twice continuously
differentiable, $\sigma(x)=\sigma(-x)>0$ if $|x|< R_1$ and it is zero
for $|x|\ge R_1\,.$ Similarly, $\chi(x)=\chi(-x)>0$ if $|x|\le2R_1$ and
$\chi(x)=0$ if $|x|>R_2$ with some $R_2>2R_1\,.$ A technical condition,
$|\chi'(x)|\le K\chi(x)^{1-\ka}\,,$ where $0<\ka<1/9\,,$ will be
exploited in Lemma 2.4.
Since $w^\t_{\a,\b}=-w^\t_{\b,\a}\,,$ the condition
$F(p_\a,p_\b)=-F(p_\b,p_\a)$ of antisymmetry of $F$ clearly implies
the conservation of total momentum. For convenience, we choose
$$
F(p_\a,\,p_\b):={1\over 2}\sum_{\t=1}^d\bigl\lan G_\t(p_\a,\,p_\b),
D_{\a,\b}\bigr\ran G_\t(p_\a,\,p_\b) \hbox{ and }
X^\t_{\a,\b}\p:={1\over \sqrt 2}\bigl\lan
G_\t(p_\a,\,p_\b),D_{\a,\b}\p\bigr\ran\,,
$$
then the formal generator $\wh L$ of the random component of our
process becomes\nfootnote{The future requirements (1.5) and (1.6) imply
that ${X^\t_{\a,\b}}^*=-X^\t_{\a,\b}$,
where the adjoint is taken with respect to the measure defined by the
kinetic energy, (see[LO]).}
$$
\wh L\p={1\over 2}\sum_{\t=1}^d\sum_{\a\in I}\sum_{\b\ne\a}
\gamma_{\a,\b}(q) X^\t_{\a,\b}\bigl(X^\t_{\a,\b}\p\bigr).
\Eq (genoise)
$$
In this case the orthogonality relations
$$
\bigl\langle G_\t(p_\a,\,p_\b),\phi'(p_\a)
-\phi'(p_\b)\bigr\rangle=0
\Eq (orthorel)
$$
imply the formal conservation of energy, see {[LO]}. To have
conservation of phase volume it is also assumed that
$$
\bigl\lan D_{\a,\b},G_\t(p_\a,p_\b)\bigr\ran=0\,,
\Eq (divnull)
$$
i.e. the operators $\gamma_{\a,\b}X^\t_{\a,\b}X^\t_{\a,\b}$ are
symmetric with respect to Lebesgue measure $dp_\a dp_\b\,.$ As a
consequence we shall see that
the conservation laws imply the reversibility of Gibbs states with
respect to $\wh L\,.$ For an explicit example of $F$ and $G_\t$ see
the Appendix of [LO].
\vskip 5pt
\noindent{\bf Gibbs measures}\vskip 2pt
Let $\l=(\l_0,\l_1,\l_2,\l_3,\l_4)$ be a set of real parameters with
$\l_4>0$ and $\l^2_1+\l^2_2+\l^2_3<\underline c^2\,,$ and denote by $\Pi$
the distribution of a Poisson process of unit intensity in
$\RR^3$ . A probability measure $\PP$ on $\O$ is called
a Gibbs state for $\Cal H$ with parameters $\l$ if its conditional
distributions given the configuration outside of any cubic box
$\L\subset\RR^3$ can be represented as
$$
\PP[d\o_\L|\o_{\L^c}]=
{1\over Z_\L}\exp\left[\lambda_0|\o_\L| +
\sum_{\a=1}^n\sum_{i=1}^3\lambda_i p_\a^i
-\lambda_4{\cal H}_{\Lambda}(\omega_\Lambda,\omega_{\Lambda^c})\right]
\,\Pi(dq_\L)\,dp_\L\,,
$$
where $Z_\L$ is the normalization, and a natural decomposition
$\o_\L=(q_\L,p_\L)$ is used, see [D]. The local Hamiltonian, $\Cal
H_\L$ is defined as
$$
{\cal H}_{\Lambda}(\omega_\Lambda,\omega_{\Lambda^c})=
\sum_{q_\a\in\omega_\Lambda}\left[\phi(p_\a)+{1\over 2}
\sum_{q_\b\in\omega_\L;\;\alpha\neq\beta}V(q_\a-q_\b)
+\sum_{q_\b\in\omega_{\L^c}}V(q_\a-q_\b)\right]\,,
$$
the set of such measures will be denoted by $\Cal P_\l\,,$ see {[R]}
for the existence of Gibbs states for superstable interactions.
\vskip 5pt
\noindent{\bf Relative entropy}
\vskip 2pt
Let $Q$ and $P$ be probability measures on $\Omega$, and for any
$\L\subset\RR^3$ denote $\Cal F_\L$ the set of continuous and bounded
functions $\psi:\O\mapsto\RR$ such that $\psi(\o)=\psi(\o_\L)$ for all
$\o\in\O\,.$ The entropy of $Q$ in $\L\,,$ relative to $P\,,$ is
defined by
$$
H_\Lambda(Q|P)\ =\ \sup_{\psi\in{\cal F}_\L}\left\{\E^Q(\psi)
-\log\E^P(e^\psi)\right\}
\Eq (entropy)
$$
where $\E^Q$ denotes the expectation with respect to the probability
measure $Q\,.$ If $\L=\RR^3$ then the subscript $\L$ of $H_\L$ will be
omitted, for properties of $H_\L$ see, for example, [OVY]. As a
reference measure a distinguished, translation invariant, Gibbs state
$P=\PP$ will be chosen. We say that $Q$ has finite specific entropy if
there exists a constant $C$ such that $H_\L(Q|\PP)\le C(1+|\L|)$ for any
cubic box $\L\,.$ If $Q$ is translation invariant with finite specific
entropy, then the particle density $\rho=\rho(\o)$ is $Q$-a.s. defined
as the following limit taken along any increasing sequence of cubic
boxes, see [LO],
$$
\rho(\o)=\lim_{|\Lambda|\to\infty}|\Lambda|^{-1}|\omega_\Lambda| .
$$
Main results of [LO] for the system under considerations can be
summarized as follows.
\proclaim {\Theorem (old)}.
Suppose that $Q$ is a translation invariant probability measure on
$\Omega$ with finite specific entropy, and let $\rho_c:=3/(4\pi
R^3_1)\,.$ If
\item{(i)} $Q[\rho(\omega)>\rho']\;=\; 1$ for some $\rho'>\rho_c\,,$
\item{(ii)} $Q$ is invariant with respect to $\wt L=L+\wh L$
in the sense that, for any smooth local function $\psi$ we have
$\E^Q(\wt L\psi)=0\,,$
\item{(iii)} $Q$ is reversible with respect to $\wh L\,,$
i.e. $\E^Q(\psi\wh L\p)\ =\ \E^Q(\p\wh L\psi)\;$ for any pair
$\p,\psi$ of smooth local functions,
\vskip-\medskipamount
\noindent{\sl then Q is a convex combination of Gibbs states.}
\medskip
\vskip 5pt
\noindent{\bf Statement of the result}
\vskip 2pt
Notice that the theorem above is stated without any reference to the
existence of the infinite dynamics, properties (ii) and (iii) of
invariance are purely formal. However, the extraction of local
information
as reversibility is usually based on a method of Liapunov functions,
namely entropy and its rate of change are compared, so the first step
of our argument is intrinsically related to the evolution.
\proclaim {\Theorem (dynamics)}. Under the conditions on the
stochastic dynamics listed above, there exists an explicitly defined
set $\bar\O\subset\O$ such that $Q(\bar\O)=1$ for each $Q$ with
finite specific entropy. Moreover, for each $\o_0\in\bar\O$ we have
a unique strong solution $\o(t)\,,\,t\ge0$ to (0.1) such that
$\o(0)=\o_0$ and $\o(t)\in\bar\O$ a.s. The solution is a measurable
function of the initial configuration, and every Gibbs state
$\PP\in\Cal P_\l$ with $\l_4>0$ and $\l_1=\l_2=\l_3=0$ is a stationary
measure for the random evolution.
This theorem is proven in the next section, solutions are defined by a
limiting procedure starting from finite systems. The restriction on
the parameters of a Gibbs measure in the last statement could have been
removed by elaborating some technical details, but we do not need such a
general assertion.
Having constructed the infinite evolution we can
consider stationary measures instead of simply measures formally invariant
as in theorem 1.1 (ii).\nfootnote{Notice that, since the infinite dynamics
satisfies equations (0.1), if $Q$ is stationary, then it satisfies (ii) of
theorem 1.1.}
\proclaim {\Theorem (reversible)}. Every translation invariant
stationary measure with finite specific entropy is reversible with
respect to the stochastic part $\wh L$ of the generator; that is,
condition (iii) in Theorem 1.1 holds.
The proof of Theorem \equ (reversible) is the content of Section 3.
Combining the above results we get the final result of the paper:
\proclaim {\Theorem (main)}. Let $Q$ be a translation invariant
stationary
measure with finite specific entropy, then condition (i) of Theorem 1.1
implies that $Q$ is a superposition of Gibbs states.
\vskip .5cm
{\bf 2. INFINITE DYNAMICS}
\vskip.25cm
\numsec=2\numfor=1\numtheo=1
We start this section by describing the set of allowed
initial configurations. Although the definition is a bit technical our
choice boils down to configurations for which the energy in a box does not
grow too fast with respect to the size of the box. The exact meaning
of this construction will become
more clear later on when the desired a priori bounds for a family of
partial dynamics and the requirements for the existence of a unique
limiting dynamics are discussed.
\vskip 5pt
\noindent{\bf Initial conditions}\vskip 2pt
Let $\Cal H_m(\o,r)$ denote the total energy of $\o\in\Omega$ in a ball
$B_m(r)\subset
\RR^3$ of center $m$ and radius $r\le\infty$, i.e. $\Cal H_m(\o,r)
\equiv \Cal H(\o_{B_m(r)})$; the number of points of $q$ in
$B_{q_\a}(r)$ will be denoted as $N_\a(q,r)\,.$ For $\kappa\in
(0,1/9)\,,$ see \equ(gamma), and $r\ge R_3 = R_1+R_2$, define
$$
\bar \Cal H_{\kappa,r}(\o) := \sup_{|m|\le r} {\Cal H_m(\o,R_3)\over
1+|m|^{3+2\kappa}}\,,\qquad
\bar\Omega_{\kappa,r} (h) = \{\o\in\Omega \;:\; \ \bar \Cal
H_{\kappa,r}(\o)\le h \}.
$$
Let $\Cal Q_r(k)$ be the set of Borel probability measures on
$\bar\Omega_{\kappa,r} (h)$ such that $Q(\Cal H_m(\o,R_3)) \le k$
for all $|m|\le r$.
Now the set of all allowed configurations is defined as
$$
\bar\Omega_{\kappa,\infty} = \bigcup_{h>0}
\bar\Omega_{\kappa,\infty}(h)
= \{ \o\in\Omega\;:\;\bar\Cal H_{\kappa,\infty}(\o) <\infty \}.
$$
Remember that the level sets $\bar\Omega_{\kappa,\infty} (h)$ are
compact in the weak topology of $\Omega$ and, in view of
\equ(superstability), $N_\a(q,R_3)=O(\sqrt h L^{{3/2}+\ka})$ for
$q_\a\in B_m(L-R_3)$ and $(q,p)\in\bar\O_{\ka,L}(h)\,.$ We shall
see that the initial condition for the existence and uniqueness of the
limiting dynamics could have been formulated in terms of $N_\a$ only,
but a preservation of bounds on kinetic energy will be needed when we
prove locality of the dynamics.
\proclaim{\Lemma(initial)}. If $\kappa>0$ then for any fixed $k>0$ we
have
$$
\lim_{h\to\infty} \inf_{r\ge R_3} \inf_{Q\in \Cal Q_r(k)}
Q(\bar\Omega_{\kappa,r} (h) ) = 1 ,
$$
that is $Q(\bar\O_{\ka,\infty}) =1$ if $Q\in \Cal Q_\infty:=
\bigcup_{k>0}\Cal Q_\infty(k)$.
\smallskip\noi{\bf Proof:}
This statement is a direct consequence of the Markov inequality. In
fact we have some universal $v>0$ such that (by $v \ZZ^3$ we denote the
tridimensional cubic lattice of size $v$)
$$
\eqalign{
Q(\bar\O_{\ka,r}(h)^c)&\le \sum_{m\in B_0(r)\cap v\Z^3}
Q[\Cal H_m(\o,\,R_3)>vh(1+|m|^{3+2\ka})\cr
&\le \sum_{m\in B_0(r)\cap v\Z^3}{Q(\Cal H_m)\over h(1+|m|^{3+2\ka})}\le
{k\over vh}\sum_{m\in v\Z^3}{1\over (1+|m|^{3+2\ka})}
}
$$
which proves the statement for any $\kappa>0\,.$ $\qed$
\smallskip
Observe that the entropy condition $H_\L(Q|\PP)\le C (1+|\L|)$
implies $Q\in \Cal Q_\infty$ via (1.7) and (1.1), see Lemma 3.1 of
[LO].
\vskip 5pt
\noindent{\bf Local dynamics}\vskip 2pt
There are several ways to define a family of partial dynamics, the
advantage of the following construction consists in its direct relation
to Gibbs states. Let $a:\RR^3\mapsto[0,1]$ be twice continuously
differentiable with compact support, we assume also $|a'(x)|\le 1$
for all $x\in\RR^3\,.$ We interpret $a$ as a smooth
version of the indicator function of a ball, its concrete shape is not
very important. For every such cutoff $a$ and inverse temperature
$\l_4>0$ we consider a system of stochastic differential equations,
$$
\eqalign{
dq_\a=&-{1\over\l_4}e^{\l_4{\cal H}_\L(\o_\L,\o_{\L^c})}
{\partial\over\partial p_\a}\left(a(q_\a)e^{-\l_4
{\cal H}_\L(\o_\L,\o_{\L^c})}\right) dt\cr
dp_\a=& {1\over\lambda_4} e^{\lambda_4
{\cal H}_\L(\o_\L,\o_{\L^c})}{\partial\over\partial q_\a}\left(a(q_\a)
e^{-\lambda_4 {\cal H}_\L(\o_\L,\o_{\L^c})}\right)\,dt\cr
&+a(q_\a)\sum_{\b\ne\a}\gamma_{\a,\b}(q)a(q_\b)F(p_\a,p_\b)\,dt\cr
&+\sqrt{a(q_\a)}\sum_{\t=1}^d\sum_{\b\ne\a}
\sqrt{a(q_\b)\gamma_{\a,\b}(q)}G_\t(p_\a,p_\b)\,dw_{\a,\b}^\t\,,
}
\Eq(dynloc)$$
where it is assumed that $\L\subset\RR^3$ is bounded and contains the
support of $a$ in its interior; in such a situation the equations above
do not depend on the particular choice of $\L\,.$ Notice that in a
region where $a=1$ our particles follow the original equations of
motion, while they are frozen outside of the support of $a\,,$ i.e.
$\dot q_\a=\dot p_\a=0\,.$ Particles approaching the boundary of the
support of $a$ slow down, thus we have a smooth transition between
moving and frozen particles, see [F1] for a similar construction. This
means that we essentially have a finite dimensional diffusion, let
$P^t_{\lambda_4,a}$ denote the Markov semigroup induced by partial
dynamics (2.1), i.e. $P^t_{\l_4,a}\psi(\o):=\E_w(\psi(\o(t))\,,$ where
$\o(t)$ is the solution with initial condition $\o(0)=\o\,,$
$\psi:\O\mapsto\RR$ is continuous and bounded, while $\E_w$
denotes the expectation with respect to the joint distribution of our
Wiener processes.
By a direct application of the Ito lemma we see
that the (formal) generator of $P^t_{\l_4,a}$ decomposes as
$\wt L_{\l_4,a}=L_{\l_4,a}+\wh L_{a}\,,$ where
$$\eqalign{
L_{\l_4,a}\psi&=-{1\over\l_4}\sum_{\a\in I}
e^{\l_4{\cal H}_\L(\o_\L,\o_{\L^c})}{\partial\over\partial p_\a}
\left(a(q_\a)e^{-\l_4 {\cal H}_\L(\o_\L,\o_{\Lambda^c})}\right)
{\partial\psi\over\partial q_\a}\cr
&+{1\over\l_4}\sum_{\a\in I} e^{\lambda_4{\cal H}_\L
(\o_\L,\o_{\L^c})}{\partial\over\partial q_\a}
\left(a(q_\a)e^{-\l_4 {\cal H}_\L(\o_\L,\o_{\L^c})}\right)
{\partial\psi\over\partial p_\a} ,\cr
\wh L_{a}\psi &={1\over 2}\sum_{\t=1}^d\sum_{\a\in I}\sum_{\b\ne\a}
\gamma_{\a,\b}(q)a(q_\a)a(q_\b)X^\t_{\a,\b}(X^\t_{\a,\b}\psi).
}
\Eq(pargen)$$
Since the coefficients of \equ(dynloc) are bounded smooth functions,
we have a differentiable dependence of solutions on initial
values. Therefore a class $\Cal D_a$ of twice continuously
differentiable functions forms a common core of $L_{\l_4,a}$ and
$\wh L_a\,,$ e.g. in the space
of continuous and bounded functions. An extension to the
space $L^2(\PP_\l)$ of square integrable functions with respect to a
distinguished Gibbs state $\PP_\l$ follows by the next Lemma.
\proclaim{\Lemma(pargibbs)}. Let $\l_1=\l_2=\l_3=0$ while $\l_4>0\,,$
then every Gibbs state $\PP\in\Cal P_\l$ satisfies
$$
\E^\PP(\psi_1 L_{\l_4,a}\psi_2)=-\E^\PP(\psi_2 L_{\l_4,a}\psi_1)\hbox{ and }
\E^\PP(\psi_1 \wh L_{a}\psi_2)=\E^\PP(\psi_2 \wh L_{a}\psi_1)
$$
for $\psi_1,\psi_2\in\Cal D_a\,,$ consequently $\PP$ is a stationary
measure of the process $P_{\l_4,a}^t$ for each cutoff $a$.
\smallskip\noi{\bf Proof:}
Both symmetry relations follow from the definition of $\PP$ by
integrating by parts. The property of reversibility
is a direct consequence of (1.6). Integration
by parts with respect to positions is possible because of the presence
of the cutoff $a\,.$ $\qed$\smallskip
Since (2.1) violates the law of momentum conservation in regions where
$a$ is not a constant, Lemma 2.2 is not true for general Gibbs
measures.
\vskip 5pt
\noindent{\bf Construction of the infinite dynamics}\vskip 2pt
First we derive an a priori bound for local dynamics; we show that the
initial condition is preserved for all $t>0$ and the related bound does
not depend on the particular choice of the cutoff function $a\,.$
\proclaim{\Lemma(apriori)}. There exists a constant $c_1$ depending
only on $\l_4$ and on the parameters of the infinite system (0.1) such
that
$$
\E_w\left(\Cal H_m(\o_a(t),R_3)\right)\le (c_1+c_1t)(1+\Cal H_m(\o_a(0),R_3+\bar ct)
$$
for all $m,t$ and $a\,,$ where $\o_a(t)$ is any solution to (2.1).
\smallskip\noi{\bf Proof:} Since all velocities are bounded by
$\bar c\,,$ we have $|q_\a(t)-q_\a(0)|\le \bar ct\,,$ whence
$N_\a(q(t),r)\le N_\a(q(0),r+\bar ct)\,,$ which yields an explicit
deterministic bound for the potential energy via superstability (1.1).
On the other hand,
$$
|b_\a(\o)|+\sum_{\t=1}^d\sum_{\b\ne\a}|\sigma^\t_{\a,\b}(\o)|^2
\le c_1' N_{\a}(q,R_1)\,,
$$
and the same bound holds true for the corresponding coefficients of
(2.1), from the stochastic equations by the Schwarz inequality we get
$$
\E_w\left(|p_\a(t)-p_\a(0)|^2\right)\le c_1''t(1+t) N_{\a}(q(0),R_1+\bar ct)^2\,,
\Eq(papr)$$
which completes the proof. Indeed, as $\phi(y)\le\phi(0)+\bar c|y|\,,$
taking the square root of both sides and summing for $\a\in I$ such
that $|q_\a(0)-m|\le R_3+\bar ct\,,$ we get a bound for
$\Cal H_m(\o(t),R_3)\,;$ the square of the number of points at time
zero is estimated again by superstability. $\qed$\smallskip
To prove the existence of limiting solutions when the cutoff is
removed we have to compare different partial solutions. Let $A_L$
denote
the set of twice continuously differentiable $a:\RR^3\mapsto [0,1]$
with compact support such that $|a'(x)|\le1$ for all $x$ and $a(x)=1$
if $|x|\le L\,.$ For $a,\bar a\in A_L$ let $\o(t)=(p(t),q(t))$
and $\bar\o(t)=(\bar q(t),\bar p(t))$ denote the corresponding
solutions to \equ(dynloc) with a common initial value
$\o(0)=\bar\o(0)=(\xi,\eta)\,.$ Supposing $|x|\le L-2\bar ct$, for
$|x-\xi_\a|\le R_0+2\bar ct$ we get
$\dd_t|q_\a-\bar q_\a|\le K_0|p_\a-\bar p_\a|\,,$ whence
$$
|q_\a(t)-\bar q_\a(t)|^2\le
2K_0\int_0^t |q_\a(s)-\bar q_\a(s)||p_\a(s)-\bar p_\a(s)|\,ds\,;
\Eq(qb)$$
here and in what follows, $K_0,K_1,K'_1...$ denote constants depending
only on the coefficients of the infinite system. The case of the
momentum variables is more complex. If $a=\bar a=1$ can be assumed
as before, then by Ito's formula we get
$$
\E_w\left(|p_\a(t)-\bar
p_\a(t)|^2\right)=\E_w\left(\int_0^t\bigl(J_{\a,1}(s)+J_{\a,2}(s)
+J_{\a,3}(s)\bigr)\,ds\right)\,,
$$
where
$$
\eqalign{
J_{\a,1}&=-2\sum_{\a\ne\b}\bigl\lan p_\a-\bar p_\a,
V'(q_\a-q_\b)-V'(\bar q_\a-\bar q_\b)\bigr\ran\,,\cr
J_{\a,2}&=2\sum_{\b\ne \a}\bigl\lan p_\a-\bar p_\a,
\gamma_{\a,\b}(q)F(p_\a,p_\b)-\gamma_{\a,\b}(\bar q)
F(\bar p_\a,\bar p_\b)\bigr\ran\cr
J_{\a,3}&=\sum_{\t=1}^d\sum_{\b\ne \a}
\bigl|\sqrt{\gamma_{\a,\b}(q)}G_\t(p_\a,p_\b)
-\sqrt{\gamma_{\a\b}(\bar q)}G_\theta(\bar p_\a,\bar p_\b)\bigr|^2 .
}
$$
Introduce now $\Ga_i$ for $i=1,2$ and $r,t\ge0$ by
$$
\Ga_1(\xi,\eta,a,\bar a;r,t):=\sum_{\a:|\xi_\a|\le r}
|q_\a(t)-\bar q_a(t)|^2\,;
$$
in the definition of $\Ga_2$ the variables $q_\a$ and $\bar q_a$ should
be replaced by $p_\a$ and $\bar p_\a\,,$ respectively. Our main tool is
the following:
\proclaim{\Lemma(converg)}. Suppose that $\ka<1/9\,,$
$2\bar cT0$ and $i=1,2$ we have
$$
\lim_{L\to\infty}\sup_{a,\bar a\in A_L}\;\sup_{(\xi,\eta)\in\bar
\O_{\ka,L}(h)}\;\E_w\left(\Ga_i(\xi,\eta,a,\bar a;r,t)\right)=0\,,
$$
and the convergence is uniform on the time interval $[0,\,T]$.
\smallskip\noi{\bf Proof:} The idea of the proof is to define and to
estimate a ``distance" (based on $\Gamma_i$)
among different partial dynamics in boxes of radius $r0$ implies $|\ti q^{\a,\b}_{\a}
-\ti q^{\a,\b}_{\b}|\le R_1+2\bar c t\le 2R_1\,,$ i.e. $N_\a(q,R_1)
\le N_\a(\ti q^{\a,\b},2R_1)\,.$ This means that
$N_\a(q,R_1)\le 1/\Te_{\a,\b}(\ti q^{\a,\b})\,,$ consequently
$$
\eqalign{
J_{\a,2}(t)&\le K_2'\bar N^\ka_{L,T}\ti J_{\a,1}(t)
+K'_2\ti J_{\a,2}(t)\,;\cr
\ti J_{\a,2}(t)&:=\sum_{\b\ne\a}\ga_{\a,\b}(q)
\bigl(|p_\a-\bar p_\a|^2+|p_\b-\bar p_\b|^2\bigr).
}
\Eq(pb2)
$$
In a similar way we get
$$
\eqalign{
J_{\a,3}(t)&\le K_3\ti J_{\a,2}(t)+K_3\ti J_{\a,3}(t)\cr
&+K_3\sum_{\b\ne\a}{\si(q_\a-q_\b)\over\Te_{\a,\b}(\ti q^{\a,\b})}
\Bigl(\sum_{\de\in I}|\dd_\de \Te_{\a,\b}(\ti q^{\a,\b})|
|q_\de-\bar q_\de|\Bigr)^2\,;\cr
&\ti J_{\a,3}(t):=\sum_{\de\in I}\ti\ga_{\a,\b}(t)
\bigl(|q_\a-\bar q_\a|^2+|p_\de-\bar p_\de|^2\bigr)
}
$$
whence by the Cauchy inequality and \equ(dga)
$$
J_{\a,3}(t)\le K'_3\ti J_{\a,2}+K'_3\bar N^{2\ka}_{L,T}\ti J_{\a,3}(t) .
\Eq(pb3)
$$
Introduce now $d(r,t):=\E_w\left(\Ga_2(\xi,\eta,a,\bar a;r,t)\right)
+\bar N_{L,T}\E_w\left(\Ga_1(\xi,\eta,a,\bar a;r,t)\right)$ for $t0$ depending only on $R_3$ and $T$, while $\bar N_{L,T}=
O(\sqrt hL^{3/2+\ka})\,.$ Using $|q_\a(t)-\xi_\a|\le \bar ct$ and the
second a priori bound \equ(papr), we see that the right hand side of
\equ(expl) vanishes as $L\to+\infty$ because
$\ell!=O\bigl((\ell/e)^\ell\bigr)$ and $({1/2}+\ka)({3/2}+\ka)<1$ by
hypothesis.
$\qed$\smallskip
Now we are in a position to prove the existence and uniqueness of
limiting solutions to (0.1). Let us consider a sequence of partial
solutions $\o_n=\o_n(t)\,,\,n\in\NN$ of \equ(dynloc) with a common
initial value $\o_n(0)=(\xi,\eta)\in\bar\O_{\ka,\infty}\,;$ the
corresponding cutoff $a_n:\RR^3\mapsto\RR$ is assumed to be a
decreasing smooth
function of $|x|$ such that $a_n(x)=1$ if $|x|\le n$ and $a_n(x)=0$ if
$|x|>n+1\,.$ In view of Lemma 2.4 $\o_n$ converges in
probability to some limit $\o(t)$ for each $t0\,.$ In view of Lemma 2.1
and Lemma 2.3 we know that $QP^t\in\Cal Q_{\infty}\,,$ too. While
$P^t_n$ has fairly good regularity properties, semigroup theory
does not apply directly to the limiting case. Nevertheless, all we
need in the next section is summarized as follows.
\proclaim{\Lemma(locality)}. Suppose that $\psi:\bar\O_{\ka,\infty}
\mapsto\RR$ is a continuous and bounded local function, i.e. $\psi(\o)
\equiv\psi(\o_{B_0(r)})$ for some $r>0\,,$ then
$$
\lim_{\ell\to\infty}\sup_{n>\ell+r}\sup_{Q\in\Cal Q_n(k)}
|QP^t_n\psi-QP^t\psi|=0
$$
for all $t,k>0\,,$ and the convergence is uniform on compact time intervals.
\smallskip\noi{\bf Proof:} The a priori bound of Lemma 2.3 extends
immediately to the limiting dynamics, thus for any $\e,T>0$ we have
some $\bar k>k$ and $h>\bar k$ such that
$Q(\bar\O_{\ka,r}(h))\ge1-\e\,,$
$QP^t_n(\bar\O_{\ka,r}(h))\ge1-\e\,,$
and $QP^t(\bar\O_{\ka,r}(h))\ge1-\e$ whenever $tr+R_3+2\bar cT$ and $Q\in\Cal Q_n(k)\,.$ Since
$\bar\O_{\ka,r}(\bar k)$ is compact, there exists also an $\e'>0$ such
that $|q_\a-\bar q_\a|+|p_\a-\bar p_\a|<\e'$, for all $\a\in I$ with
$|q_\a|,|\bar q_\a|\le r$, implies $|\psi(\o)-\psi(\bar\o)|\le\e$ for
$\o,\bar \o\in\bar\O_{\ka,r}(\bar k)\,.$ Therefore the statement follows
from Lemma 2.4 and Chebishev inequality by a $3\e$ argument. $\qed$\smallskip
The final statement of Theorem \equ (dynamics) on stationarity of certain Gibbs
states is now a direct consequence of Lemma 2.2.
\vskip .5cm
{\bf 3. AN ENTROPY ARGUMENT}\vskip.25cm
\numsec=3\numfor=1\numtheo=1
In this section we extend a familiar argument by Holley [H]
to the present more complex situation.
For a probability measure $Q$ on $\O$, let $H(Q|\PP_\lambda)$
denote the entropy relative to a distinguished Gibbs state
$\PP_\lambda$ with $\l_1=\l_2=\l_3=0$, as defined by \equ(entropy)
with $\Lambda = \RR^3$. The family of partial dynamics \equ(dynloc)
has been chosen such that $\PP_\l$ is a common stationary measure of
each local dynamics $P_n^t=P_{\l_4,a_n}$ introduced in Section 2.
Therefore $P_n^t$ is a strongly
continuous semigroup in $L^2(\PP_\lambda)\,,$ and smooth cylinder
functions form a core for its generator $\wt L_n = L_n +\wh L_n\,,$
see \equ(pargen). Remember that $L_n\equiv L_{\l_4,a_n}$, the
Hamiltonian part, is antisymmetric in $L^2(\PP_\l)$ while the symmetric
(reversible) component is just $\wh L_n\equiv \wh L_{a_n}\,.$
If $\G$ is a generator in $L^2(\PP_\l)$ then the corresponding
Donsker-Varadhan rate function is defined as
$$
D(Q|\G) \ =\ \sup_\psi\Bigl\{ - \int {\G \psi\over \psi} dQ \;:\;
\psi\in \hbox{Dom }\G,\,\inf \psi > 0\Bigr\} .
$$
If $\G$ is self-adjoint and $\G<0$, then we can apply the following
result due to Donsker and Varadhan (cf. [DV], Theorem 5)
\proclaim{\Theorem (donsker-var)}.
$D (Q|\G )<+\infty$ if and only if $Q<<\PP_\l$ and
$g:=\sqrt{dQ/d\PP_\l}\in\hbox{Dom }\sqrt{-\G}$; moreover
$$
D(Q|\G) \ =\ \int \left(\sqrt{-\G} g\right)^2 d\PP_\l .
\Eq(dirichlet)
$$
Our main tool consists of the following entropy inequality.
\proclaim{\PProposition (H-th)}. Let $\bar Q_n^t := (1/t) \int_0^t
QP_n^s ds$. If $H(Q|\PP_\l) <\infty$ then
$$
H(QP_n^t|\PP_\l) + 2t D(\bar Q_n^t|\hat L_n) \le H(Q|\PP_\l) .
\Eq(H-ineq)
$$
\smallskip\noi{\bf Proof:}
Let $P^{*t}_n$ be the adjoint semigroup with respect to $\PP_\l$,
which is again a diffusion with formal generator $\tilde L_n^* =
-L_n+\hat L_n$. Both forward and backward diffusion are essentially
finite dimensional with smooth coefficients, thus twice continuously
differentiable functions form a common core $\Cal D_n$ of $L_n$ and
$L^*_n\,.$ This suffice to justify the following computations.
Observe first that, as an easy consequence of Jensen's inequality, we
have
$$
H(Q'P_n^{\tau}|Q''P_n^{\tau}) \le H(Q'|Q'')
\Eq (H-increase)
$$
for any two measures $Q',\,Q''$.
For any strictly positive $\psi\in\Cal D_n$ with $\PP_\l(\psi)=1$
define $Q''$ by $dQ''=\psi d\PP_\l\,.$ Since
$$
{dQ''P_n^{\tau}\over d\PP_\l} = P^{*{\tau}}_n \psi
$$
we have
$$
H(Q'|Q'') = H(Q'|\PP_\l) - Q'(\log \psi)
$$
and
$$
H( Q' P_n^{\tau} | Q''P_n^{\tau}) = H(Q'P_n^{\tau}|\PP_\l)
- Q'P_n^{\tau}(\log P^{*{\tau}}_n\psi).
$$
Accordingly, by \equ(H-increase),
$$
H(Q'|\PP_\l) - H(Q'P_n^{\tau}|\PP_\l) \ge Q'(\log \psi)
- Q'P_n^{\tau}(\log P^{*{\tau}}_n\psi),
$$
whence, by the concavity of the logarithm and the inequality
$\log(x+1) \le x$,
$$
H(Q'|\PP_\l) - H(Q'P_n^{\tau}|\PP_\l) \ge Q'(\log \psi) -
Q'(\log P_n^{\tau} P^{*{\tau}}_n\psi)\ge
\int {\psi - P_n^{\tau} P^{*{\tau}}_n\psi\over \psi} dQ' .
$$
Remembering that $P_n^t$ and $P^{*t}_n$ are both Feller semigroup and $\psi$
belongs to the common core of $\tilde L_n$ and $\tilde L_n^*$,
we have, for small $\tau$,
$$
\psi - P_n^{\tau} P^{*\tau}_n\psi
= \psi - P_n^{\tau} \psi+ \psi - P^{*\tau}_n\psi
+ P_n^{\tau}\left(\psi - P^{*\tau}_n\psi\right)-
\left(\psi - P^{*\tau}_n\psi\right)
= -2\tau \hat L_n \psi+o(\tau).
$$
Therefore, by dividing the given interval $[0,t]$ into $m$ small pieces,
with $\tau=t/m$ and $Q'=QP_n^{{it\over m}}$, we get
$$
\eqalign{H(Q|\PP_\l) - H(QP_n^t|\PP_\l)&= \lim_{m\to\infty}
{1\over m}\sum_{i=0}^{m-1}\left[H(QP_n^{{i\over m}t}|\PP_\l)
- H(QP_n^{{i+1\over m}t}|\PP_\l)\right]\cr
&\ge -2\int_0^t ds \int {\hat L_n \psi \over \psi}\ dQP_n^s .}
$$
By taking the supremum over all $\psi$ considered we conclude the
proof. $\qed$\smallskip
Observe now that if $D (Q|\hat L_n)< \infty$, then by theorem
\equ(donsker-var)
it can be written as a sum, namely, if $g = \sqrt{dQ/d\PP_\l}$,
\nfootnote{To see this, since $g\in \hbox{Dom }\sqrt{-\hat L_n}$
(hence $g\in
\hbox{Dom }\sqrt{a(q_\a)a(q_\b)\gamma_{\a\b}(q)}X_{\a\b})$),
one can approximate it by local smooth
functions, then use the closability of the Dirichlet form $D$.}
$$
D (Q|\hat L_n ) = {1\over 2}
\int \sum_{\t, \a,\b} a_n(q_\a) a_n(q_\b) \gamma_{\a\b}(q) (X^\t_{\a\b} g)^2 \
d\PP_\l .
$$
Let $a_{n,1}(x), a_{n,2}(x),\dots, a_{n,j}(x)$ be smooth non-negative functions
with compact support, and assume that their supports are disjoint.
Furthermore, assume that $a_n(x) \ge a_{n,1}(x) + \dots + a_{n,j}(x)$,
then
$$
D (Q|\hat L_n ) \ge D (Q|\hat L_{a_{n,1}} ) + \dots + D (Q|\hat L_{a_{n,j}} ).
$$
Therefore, from \equ(H-ineq), we have
$$
H(QP_n^t|\PP_\l) + 2t \sum_{i=1}^jD(\bar Q_n^t|\hat L_{a_{n,i}}) \le
H(Q|\PP_\l).
$$
Thus we can choose strictly positive
and smooth functions $\psi_0, \psi_1,\dots,\psi_j$ such that
$$
QP_n^t (\psi_0) - \log \PP_\l(e^{\psi_0})
- 2t \sum_{i=1}^j \bar Q_n^t \left({\hat L_{a_{n,i}}\psi_i\over \psi_i}\right)
\le H(Q|\PP_\l) .
$$
This inequality extends by continuity to the infinite dynamics
(cf. lemma \equ(locality) and note that $Q\in \Cal Q_\infty$)
$$
QP^t (\psi_0) - \log \PP_\l(e^{\psi_0})
- 2t \sum_{i=1}^j \bar Q^t \left({\hat L_{a_{n,i}}\psi_i\over \psi_i}\right)
\le H(Q|\PP_\l) .
\Eq(ready)
$$
Now we are in a position to take the thermodynamic limit and conclude
the main result of this section.
\proclaim{\PProposition (loc-sta)}.
If $Q_*$ is a translation invariant stationary measure of the
infinite system \equ(stochdyn), and $Q_*$ has finite specific
entropy
with respect to $\PP_\l$, then $D( Q_*|\wh L_{\bar a}) = 0$ for all smooth
functions $\bar a\le 1$ of compact support.
\smallskip\noi{\bf Proof}:
We are going use \equ(ready) with $Q=Q_{*n}$ where $Q_{*n}$
is defined by
$$
Q_{*n}(\psi)\ =\ \int \PP_\l(\psi|\Cal F_{\L_n})\ dQ_*
$$
and $\L_n$ denotes the centered cubic box of size $n$.
Of course, $H(Q_{*n}|\PP_\l) = H_{\L_n}(Q_*|\PP_\l)\,,$ thus
$$
\bar H(Q_*|\PP_\l):= \lim_{n\to \infty}{H(Q_{*n}|\PP_\l)\over |\L_n|}
= \sup_{\psi}\left( Q_*(\psi) - \bar F(\psi)\right)\ ,
\Eq(H-density)
$$
where $\psi$ are the local, bounded and continuous functions; in addition,
$$
\bar F(\psi) := \lim_{n\to \infty} {1\over |\L_n|}\log \int \exp
\left(\sum_{k\in \L_n\cap \Z^3} \vec s^{k} \psi\right) d\PP_\l ,
\Eq(free)
$$
and $\vec s^{k}$ denotes the shift in $\RR^3$ by $k \in
\RR^3$, i.e. $\vec s^{k}\psi(p,q) = \psi( p,\vec s^{k}q)$ and
$\vec s^{k}q = \{q_\a +k\}$ if $q = \{q_\a\}$.
The proof of the existence of \equ(H-density) and \equ(free)
can be found in [OVY].
Now we set
$$
\psi_0 = \sum_{k\in \L_n\cap \Z^3}\vec s^{k} \psi
$$
for some local bounded continuous function $\psi$.
Without loss of generality we can suppose $n$ so large that $\L_n$ contains the
support of $\bar a$, and
define $a_{n,i}(x) = \bar a (x+k_i)$,
$k_i \in J_n$, and $J_n$ is a discrete subset of $\L_n$
such that the $a_{n,i}$
have the disjoint supports contained in $\L_n$, and ${|J_n|\over n^3 }\ge
\bar J_0$, for some fixed constant $\bar J_0$.\nfootnote{This can be
done as to ensure that $a_n\geq\sum_{i=1}^ja_{n,i}$.}
Correspondingly we choose $\psi_i =\vec s^{k_i} \bar\psi,\ \vec
k_i\in J_n$,
for a local bounded continuous function $\bar\psi$.
Substituting in \equ(ready) and dividing by $|\L_n|$,
it remains to prove that
$$
\lim_{n\to \infty} {1\over |\L_n|}
\sum_{k\in \L_n\cap \Z^3} Q_{*n} P^t(\vec s^{k} \psi) =
Q_*(\psi)
\Eq(unif1)
$$
and
$$
\lim_{n\to \infty} {1\over |J_n|} \sum_{k_i\in J_n}
\bar Q_{*n}^{t} \left(\vec s^{k_i}{\wh L_{\bar a}\bar\psi
\over\bar \psi}\right)
= Q_* \left({\wh L_{\bar a}\bar\psi\over \bar\psi} \right) .
\Eq(unif2)
$$
Indeed, then \equ(ready), \equ(H-density),
\equ(free), \equ(unif1) and \equ(unif2)
imply
$$
Q_* (\psi) - \bar F(\psi)
- 2t J_0 Q_* \left({\hat L_{\bar a}\bar\psi\over\bar\psi}\right)
\le \bar H(Q_*|\PP_\l),
$$
and taking the supremum over all $\psi$ and $\bar\psi$ considered
we obtain the wanted result.
To prove \equ(unif1), observe first that the rate of
convergence in Lemma 2.5 depends only on the magnitude and the
modulus of continuity of the underlying function. In the present
situation all functions are translates of each other, thus the convergence
is uniform on such functions. Therefore, for $k\in\L_{n-\sqrt n}$ we
approximate $P^t$ with the local dynamics $\vec s^kP^t_{\sqrt n}$
in the ball $B_k(\sqrt n)\,,$ otherwise we use simply the uniform bound
of $\psi\,.$ The proof of \equ(unif2) is similar.$\qed$
As it is well known, see [DV], $D(Q_*|\wh L_{\bar a})=0$ implies the
reversibility of $Q_*$ with respect to $\wh L_{\bar a}\,,$ which
completes the proof of Theorem 1.3, whereby proving Theorem 1.4 as well,
by a direct argument.
\vskip .5cm
{\bf REFERENCES.}\vskip.25cm
\frenchspacing
\item{[BLPS]}
L. Bunimovich, C. Liverani, A. Pellegrinotti, Y. Suhov: Ergodic Systems
of n Balls in a Billiard Table. Commun. Math. Phys.
{\bf 146} (1992), 357--396.
\item{[D]}
R.L. Dobrushin: Gibbsian random fields for particles without hard core.
(in Russian) Teor. Mat. Fiz. {\bf 4} (1969), 101--118.
\item{[DL]}
V. Donnay, C. Liverani: Potential on the Two-Torus for which the
Hamiltonian Flow is ergodic. Commun. Math. Phys. {\bf 135} (1991),
267--302.
\item{[DV]}
M.D. Donsker, S.R.S. Varadhan: Asymptotic Evaluation of Certain Markov
Process
Expectations for Large Time. I, Commun. Math. Phys. {\bf 28} (1975),
1--47.
\item{[FD]}
J. Fritz, R.L. Dobrushin: Non-equilibrium dynamics of
two-dimensional infinite particle systems with a singular interaction.
Commun. Math. Phys. {\bf 57} (1977), 67--81.
\item{[FFL]}
J. Fritz, T. Funaki, J.L. Lebowitz: Stationary States of Random
Hamiltonian Systems. Probab. Theory Related Fields {\bf 99} (1994),
211--236.
\item{[F1]}
J. Fritz: Gradient dynamics of infinite point systems. Ann. Prob.
{bf 15} (1987), 478--514.
\item{[F2]}
J. Fritz: Stationary States of Hamiltonian Systems with Noise. In:
On Three Levels, 203--214, M. Fannes, Ch. Maes, A. Verbeure Eds,
Plenum, New York, 1994.
\item{[H]} R. Holley: Free energy in a Markovian model of a lattice
spin system. Commun. Math. Phys. {\bf 23} (1971), 87-99.
\item{[KSS]}
A. Kr\'amli, N. Simanyi, D. Sz\'asz: The K property for Four
Billiard Balls. Commun. Math. Phys. {\bf 144} (1992), 107--148.
\item{[LO]}
C. Liverani, S. Olla: Ergodicity in Infinite Hamiltonian
Systems with Conservative Noise. Probab. Theory Related Fields, {\bf 106},
3, (1996), 401--445.
\item{[LW]}
C. Liverani, M. Wojtkowski: Ergodicity in Hamiltonian Systems, Dynamics
Reported {\bf 4} (1995), 130--202.
\item{[OVY]}
S. Olla, S.R.S. Varadhan, H. T. Yau : Hydrodynamics Limit for a
Hamiltonian
System with Weak Noise. Commun. Math. Phys. {\bf 155} (1993), 523--560.
\item{[R]}
D. Ruelle: Superstable interactions in classical statistical mechanics,
Commun. Math. Phys. {\bf 18} (1970), 127-159.
\item{[S]}
R. Siegmund-Schultze: On nonequilibrium dynamics of
multidimensional infinite particle systems. Commun. Math. Phys. {\bf
100} (1985), 245-265.
\bye
ENDBODY